Let \( \mathcal {H}(\mathbb {D}) \)  be the class of analytic functions \( f(z)=\sum _{n=0}^{\infty }a_nz^n \)  in the unit disk \( \mathbb {D}\) . The origin of the Bohr phenomenon lies in the seminal work by Harald Bohr [1]}, which include the following result: Let \( f\in \mathcal {H}(\mathbb {D}) \)  satisfies \( |f(z)|<1 \)  for all \( z\in \mathbb {D} \) , then \( \sum _{n=0}^{\infty }|a_n||z|^n\le 1 \)  for all \( z\in \mathbb {D} \)  with \( |z|=r\le 1/3 \)  and the constant \( 1/3 \)  is the best possible. For \( f\in \mathcal {H}(\mathbb {D}) \) , the majorant series is denoted by \( M_f(r) \)  and is defined by \( M_f(r)=\sum _{n=0}^{\infty }|a_n||z|^n \) . Bohr actually obtained the inequality \( M_f(r)\le 1 \)  for \( |z|\le 1/6 \) , but subsequently later, M. Riesz, I. Schur and F. Weiner, independently established this inequality for \( |z|\le 1/3 \)  and the constant \( 1/3 \)  cannot be improved [2]}. The constant \( r_0=1/3 \)  is called the Bohr radius and the inequality \( M_f(r)\le 1 \)  is called Bohr inequality for bounded analytic functions in the unit disk \( \mathbb {D}. \)  Moreover, for the function \( \phi _a \)  defined by
\(\phi _a(z)=\frac{a-z}{1-az},\;\; a\in [0,1)\) 
